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Round-off Effects in FIR Digital Filters 587
bins = [-K+0.5:1:K-0.5]; % Bin values from -K to K
Q = bins/DbM; % Normalized bins
YTN = 2^(-bM);
% Ytick marks interval
YLM = 20*YTN;
% Yaxis limit
% Quantize the input and the filter coefficients
h1 = QFix(h1,B,’round’,’satur’); % h1 quantized to B bits
h2 = QFix(h2,B,’round’,’satur’); % h1 quantized to B bits
xn = QFix(xn,B,’round’,’satur’); % Input quantized to B bits
% Filter output without multiplication quantization
yn = casfiltr(b0,Bh,Ah,xn); % output using Casfiltr routine
% Filter output with multi quant (1 multiplier/section)
xq = QFix(xn,B,’round’,’satur’); % Section-1 scaled input
wn = filter(h1,1,xq);
% Sec-1 unquantized output
wq = QFix(wn,B,’round’,’satur’); % Sec-1 quantized output
wq = QFix(wq,B,’round’,’satur’); % Section-2 scaled input
yq = filter(h2,1,wq);
% Sec-2 unquantized output
yq = QFix(yq,B,’round’,’satur’); % Sec-2 quantized output
% Output Error Analysis
qn = yn-yq;
% Outout error sequence
varyn = var(yn); varqn = var(qn); % Signal and noise power
qqmax = max(qn); qqmin = min(qn); % Maximun and minimum of the error
qnmax = max(abs([qqmax,qqmin])); % Absolute maximum range of the error
qnavg = mean(qn); qnstd = std(qn); % Mean and std dev of the error
qn = round(qn*(2^bM)/(2*qnmax)+0.5); % Normalized en (interger between -K & K)
qn = sort([qn,-K:1:(K+1)]); %
H = diff(find(diff(qn)))-1; % Error histogram
H = H/N;
% Normalized histogram
Hmax = max(H); Hmin = min(H);
% Max and Min of the normalized histogram
% Output SNRs
SNR_C = 10*log10(varyn/varqn); % Computed SNR
SNR_T = 6.02*(B+1) + 10*log10(sum(h.*h)) ...
- 10*log10(1+sum(h2.*h2)); % Theoretical SNR
The plot is shown in Figure 10.29. The error distribution appears to have a
Gaussian envelope, but the error is not continuously distributed. This behavior
indicates that the output error takes only a fixed set of values, which is due
to a particular set of coefficient values. The computed SNR is 70.85 dB,
which agrees with the above theoretical value. Thus, our assumptions are
reasonable.
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10.3.4 FLOATING-POINT ARITHMETIC
Analysis for the floating-point arithmetic is more complicated and tedious.
Hence we will consider only the direct-form realization with simplified
assumptions. Figure 10.30 shows a direct-form realization with a floatingpoint
arithmetic model. In this realization, {η i (n)}, 1≤ i ≤ M − 1 are
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